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We study the class of virtually homo-uniserial modules and rings as a nontrivial generalization of homo-uniserial modules and rings. An R-module M is virtually homo-uniserial if, for any finitely g...

We study the class of virtually uniserial modules and rings as a nontrivial generalization of uniserial modules and rings. An R-module M is virtually uniserial if for every finitely generated submodule 0 ≠ K ⊆ M , K / Rad ( K ) is virtually simple. Also, an R-module M is called virtually serial if it is a direct sum of virtually uniserial modules and a left virtually uniserial (resp., left virtually serial) ring is a ring which is virtually uniserial (resp., serial) as a left R-module. We give some useful properties of virtually (uni)serial modules and rings. In particular, it is shown that every left virtually uniserial module is uniform and Bezout. Also, we show that if R is a left virtually serial ring, then R / J ( R ) ≅ ∏ i = 1 t M n i ( D i ) where t , n 1 , … , n t ∈ N and each D i is a principal left ideal domain. As a consequence, we obtain that a ring R is left virtually serial with J ( R ) = 0 if and only if R ≅ ∏ i = 1 t M n i ( D i ) where t , n 1 , … , n t ∈ N and each D i is a principal left ideal domain with J ( D i ) = 0 . Also, several classes of rings for which every virtually uniserial module (resp., ring) is uniserial are given. Noetherian left virtually uniserial rings are characterized. Finally, we obtain some structure theorems for (commutative) rings over which every (finitely generated) module is virtually serial.

A famous theorem of commutative algebra due to I. M. Isaacs states that “if every prime ideal of R is principal, then every ideal of R is principal”. Therefore, a natural question of this sort is “whether the same is true if one weakens this condition and studies rings in which ideals are direct sums of cyclically presented modules?” The goal of this paper is to answer this question in the case R is a commutative local ring. We obtain an analogue of Isaacs's theorem. In fact, we give two criteria to check whether every ideal of a commutative local ring R is a direct sum of cyclically presented modules, it suffices to test only the prime ideals or structure of the maximal ideal of R. As a consequence, we obtain: if R is a commutative local ring such that every prime ideal of R is a direct sum of cyclically presented R-modules, then R is a Noetherian ring . Finally, we describe the ideal structure of commutative local rings in which every ideal of R is a direct sum of cyclically presented R-modules.

A ring R is said to be left pure-hereditary (resp. RD-hereditary) if every left ideal of R is pure-projective (resp. RD-projective). In this paper, some properties and examples of these rings, which are nontrivial generalizations of hereditary rings, are given. For instance, we show that if R is a left RD-hereditary left nonsingular ring, then R is left Noetherian if and only if u . dim ( R R ) ∞ . Also, we show that a ring R is quasi-Frobenius if and only if R is a left FGF, left coherent right pure-injective ring. A ring R is said to be left FP-hereditary if every left ideal of R is FP-projective. It is shown that if R is a left CF ring, then R is left Noetherian if and only if R is left pure-hereditary, if and only if R is left FP-hereditary, if and only if R is left coherent. It is shown that every left self-injective left FP-hereditary ring is semiperfect. Finally, it is shown that a ring R is left FP-hereditary (resp. left coherent) if and only if every submodule (resp. finitely generated submodule) of a projective left R-module is FP-projective, if and only if every pure factor module of an injective left R-module is injective (resp. FP-injective), if and only if for each FP-injective left R-module U, E ( U ) / U is injective (resp. FP-injective).

We study direct-sum decompositions of pure-projective modules over some classes of rings. In particular, we determine several classes of rings over which every pure-projective left module is a direct sum of cyclic modules. Finally, the relationship between our results and FGC rings is also studied.

A famous theorem of algebra due to Osofsky states that “if every cyclic left R -module is injective, then R is semisimple”. Therefore, a natural question of this sort is: “What is the class of rings R for which every cyclic left R -module is pure-injective or pure-projective?” The goal of this paper is to answer this question. For instance, we show that if every cyclic left R -module is pure-injective, then R is a left perfect ring. As a consequence, a commutative coherent ring R is Artinian if and only if every cyclic R -module is pure-injective. Also, a commutative ring R is pure-semisimple (i.e., every R -module is pure-injective) if and only if all cyclic R -modules and all indecomposable R -modules are pure-injective. We obtain some generalizations of Osofsky's theorem in the cases R is semiprimitive or commutative coherent or a commutative semiprime Goldie ring. Finally, we show that a ring R is left Noetherian if and only if every cyclic left R -module is pure-projective. As a corollary of this result we obtain: if every cyclic left R -module is pure-injective and pure-projective, then R is a left Artinian ring. The converse is also true when R is commutative.

The statement of Theorem 2.1 in [4] is not correct. We give a counterexample to this statement and following to this error we give further corrections. We also offer two better substitutions for Proposition 2.9 and Theorem 2.11 in [4] .

In this paper, we obtain a partial solution to the following question of Köthe: For which rings R R is it true that every left (or both left and right) R R -module is a direct sum of cyclic modules? Let R R be a ring in which all idempotents are central. We prove that if R R is a left Köthe ring (i.e., every left R R -module is a direct sum of cyclic modules), then R R is an Artinian principal right ideal ring. Consequently, R R is a Köthe ring (i.e., each left and each right R R -module is a direct sum of cyclic modules) if and only if R R is an Artinian principal ideal ring. This is a generalization of a Köthe-Cohen-Kaplansky theorem.

We study direct-sum decompositions of RD -projective modules. In particular, we investigate the rings over which every RD -projective right module is a direct sum of cyclically presented right modules, or a direct sum of finitely presented cyclic right modules, or a direct sum of right modules with local endomorphism rings (SSP rings). SSP rings are necessarily semiperfect. For instance, the superlocal rings introduced by Puninski, Prest and Rothmaler in [28] and the semilocal strongly π -regular rings introduced by Kaplansky in [24] are SSP rings. In the case of a Noetherian ring R (with further additional hypotheses), an RD -projective R -module M turns out to be either a direct sum of finitely presented cyclic modules or of the form M = T ( M ) ⊕ P , where T ( M ) is the torsion part of M (elements of M annihilated by a regular element of R ) and P is a projective module.

In this article, several characterizations of certain classes of rings via FC-purity and I-purity are considered. Among others results, it is shown that every I-pure injective left R-module is projective if and only if every FC-pure projective left R-module is injective, if and only if, R is a semisimple ring. In particular, the structures of FC-pure projective and I-pure projective modules over a left Artinian ring are completely described. Also, it is shown that every left R-module is FC-pure projective if and only if every indecomposable left R-module is a finitely presented cyclic R-module, if and only if, R is a left Kothe ring. Finally, we introduce FC-pure flatness and I-pure flatness of modules and several characterizations of these notions are given. In particular, we show that a commutative ring R is quasi-Frobenius if and only if R is an Artinian ring and I-pure injective, if and only if, R is an Artinian ring and the injective envelope E(R) is an FC-pure projective R-module.

A left [Formula: see text]-module [Formula: see text] is said to be left singly injective if [Formula: see text] for any cyclic submodule [Formula: see text] of any finitely generated free left [Formula: see text]-module [Formula: see text]. In this paper, we study the notion of singly injective modules which is generalization of injective modules and absolutely pure modules. In this direction, we give conditions which guarantee that each singly injective left [Formula: see text]-module is either injective or absolutely pure. Finally, we study rings whose simple modules are singly injective (SSI-rings).

This paper investigates the structure of cyclically pure (or C-pure) projective modules. In particular, it is shown that a ring R is left Noetherian if and only if every C-pure projective left R-module is pure projective. Also, over a left hereditary Noetherian ring R, a left R-module M is C-pure projective if and only if M = N ⊕ P, where N is a direct sum of cyclic modules and P is a projective left R-module. The relationship C-purity with purity and RD-purity are also studied. It is shown that if R is a local duo-ring, then the C-pure projective left R-modules and the pure projective left R-modules coincide if and only if R is a principal ideal ring. If R is a left perfect duo-ring, then the C-pure projective left R-modules and the pure projective left R-modules coincide if and only if R is left Kothe (i.e., every left R-module is a direct sum of cyclic modules). Also, it is shown that for a ring R, if every C-pure projective left R-module is RD-projective, then R is left Noetherian, every p-injective l...

A theorem from commutative algebra due to Kothe and Cohen-Kaplansky states that, “a commutative ring R has the property that every R -module is a direct sum of cyclic modules if and only if R is an Artinian principal ideal ring”. Therefore, an interesting natural question of this sort is “whether the same is true if one only assumes that every ideal is a direct sum of cyclic modules?” The goal of this paper is to answer this question in the case R is a finite direct product of commutative Noetherian local rings. The structure of such rings is completely described. In particular, this yields characterizations of all commutative Artinian rings with this property.